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Gelfond's method for algebraic independence


Author: W. Dale Brownawell
Journal: Trans. Amer. Math. Soc. 210 (1975), 1-26
MSC: Primary 10F35
DOI: https://doi.org/10.1090/S0002-9947-1975-0382181-1
MathSciNet review: 0382181
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Abstract: This paper extends Gelfond's method for algebraic independence to fields $ K$ with transcendence type $ \leqslant \tau $. The main results show that the elements of a transcendence basis for $ K$ and at least two more numbers from a prescribed set are algebraically independent over $ Q$. The theorems have a common hypothesis: $ \{ {\alpha _1}, \ldots ,{\alpha _M}\} ,\{ {\beta _1}, \ldots ,{\beta _N}\} $ are sets of complex numbers, each of which is $ Q$-linearly independent.

THEOREM A. If $ (2\tau - 1) < MN$, then at least two of the numbers $ {\alpha _i},{\beta _j},\exp ({\alpha _i}{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N$, are algebraically dependent over $ K$.

THEOREM B. If $ 2\tau (M + N) \leqslant MN + M$, then at least two of the numbers $ {\alpha _i},\exp ({\alpha _i},{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N$, are algebraically dependent over $ K$.

THEOREM C. If $ 2\tau (M + N) \leqslant MN$, then at least two of the numbers $ 1 \leqslant i \leqslant M,1 \leqslant j \leqslant N$, are algebraically dependent over $ K$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0382181-1
Keywords: Transcendental numbers, algebraically independent numbers, exponential function, diophantine approximations, sequences of approximations, Liouville estimates, transcendence measure, transcendence type
Article copyright: © Copyright 1975 American Mathematical Society

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